Applicability of the Bramble-Hilbert lemma in interpolation problems of narrow quadrilateral isoparametric finite elements

Two approaches to deriving the interpolation theorem for narrow quadrilateral isoparametric finite elements not satisfying the condition ρKhK ⩾ ρ0 > 0, ρK and hK being the radius of inscribed circle and the diameter of the quadrilateral K, respectively, are presented. The first one, using the Bramble-Hilbert lemma, is successful only in deriving the L2(K)-estimate. The nonapplicability of the standard approach via the Bramble-Hilbert lemma in the case of H1(K)-estimate is presented and a fully efficient method giving the optimum rate of convergence O(h) in the H1(K)-norm is described. In the end, the dependence of the interpolation error on the geometry of a quadrilateral is demonstrated by an example.